Problem: Yasemin deposited $\$1000$ into a savings account. The relationship between the time, $t$, in years, since the account was first opened, and Yasemin's account balance, $B(t)$, in dollars, is modeled by the following function. $B(t)=1000 \cdot e^{0.03t}$ How many years will it take for Yasemin's account balance to reach $\$1500$ ? Round your answer, if necessary, to the nearest hundredth.
Solution: Thinking about the problem We want to know how many years, $t$, it will take for Yasemin's account balance, $B(t)$, to reach $\$1500$. So we need to find the value of $t$ for which $B(t)=1500$. Substituting $1500$ in for $B(t)$ in the function gives us the following equation. $1500=1000 \cdot e^{0.03t}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}1000\cdot e^{0.03t}&=1500\\\\ e^{0.03t}&=1.5\\\\ 0.03t&=\ln(1.5)\\\\ t&=\dfrac{\ln(1.5)}{0.03}\\\\ t&\approx 13.52\end{aligned}$ Yasemin's account balance will reach $\$1500$ after $13.52$ years.